Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Represent the complex number graphically, and find the standard form of the number. $$\frac{1}{4}\left[\cos \left(-45^{\circ}\right)+i \sin \left(-45^{\circ}\right)\right]$$

Step-by-Step Solution

Answer

The standard form of the complex number is \( 1/(4\sqrt{2}) - i/(4\sqrt{2}) \), and it's represented by the point \( (1/(4\sqrt{2}), -1/(4\sqrt{2})) \) on the complex plane.

Step 1: Identify the magnitude and angle in the given polar form

In the given complex number, \( r = 1/4 \) and \( θ = -45^\circ \) or \( -\pi/4 \) in radians.

Step 2: Calculate the real and imaginary parts of the complex number

Use the polar form to calculate the real and imaginary parts of the complex number. The real part is given by \( r \cos θ \) and the imaginary part is given by \( r \sin θ \). Substituting the given values, we get \[ Real = 1/4 \cos (-\pi/4) = 1/(4\sqrt{2}) \] and \[ Imaginary = 1/4 \sin (-\pi/4) = -1/(4\sqrt{2}) \]

Step 3: Write the standard form of the complex number

The standard form of a complex number is \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. Substituting our calculated values, we get the standard form as \( 1/(4\sqrt{2}) - i/(4\sqrt{2}) \).

Step 4: Plot the complex number

Plot the complex number on the complex plane. The x-axis represents the real part and the y-axis represents the imaginary part. The complex number, therefore, is located \( 1/(4\sqrt{2}) \) units to the right and \( 1/(4\sqrt{2}) \) units down from the origin.

Key Concepts

Polar Form

The polar form of a complex number allows us to express it in terms of its magnitude and angle. It is a particularly useful way to represent complex numbers when dealing with multiplication and division.
The polar form is written as \[ r \left(\cos \theta + i \sin \theta \right) \] where \( r \) is the magnitude or modulus of the complex number, and \( \theta \) is the angle made with the positive real axis, known as the argument.
In our example, the given complex number is \[ \frac{1}{4}\left[\cos \left(-45^{\circ}\right)+i \sin \left(-45^{\circ}\right)\right] \].Here, \( r = \frac{1}{4} \) and \( \theta = -45^{\circ} \) (or \(-\pi/4\) radians).

• Magnitude (r): This tells us the distance from the origin to the point on the complex plane.
• Angle (\(\theta\)): This shows the direction from the positive real axis.

Standard Form

The standard form of a complex number is an alternative to the polar form. It expresses the number as \( a + bi \), where \( a \) is the real part, and \( b \) is the imaginary part.
To convert from polar to standard form, you multiply the magnitude by the cosine of the angle for the real part and by the sine of the angle for the imaginary part. Using our exercise, this means calculating:

• Real Part (a): \[ a = r \cos \theta = \frac{1}{4} \cos \left(-\pi/4\right) = \frac{1}{4\sqrt{2}} \]
• Imaginary Part (b): \[ b = r \sin \theta = \frac{1}{4} \sin \left(-\pi/4\right) = -\frac{1}{4\sqrt{2}} \]

Thus, the complex number in standard form is \( \frac{1}{4\sqrt{2}} - \frac{i}{4\sqrt{2}} \).
This form is especially useful for addition and subtraction of complex numbers.

Imaginary Numbers

Imaginary numbers are numbers that have a real number multiplied by the imaginary unit \(i\), where \(i\) is defined as the square root of \(-1\).
They are essential for representing complex numbers and often appear in engineering, physics, and mathematics.
In the standard form \( a + bi \), the imaginary number \(b\) represents the component multiplied by \(i\).

• Imaginary Unit (i): It satisfies \(i^2 = -1\).
• Imaginary Part: In our example, the imaginary part of the complex number is \(-\frac{1}{4\sqrt{2}}\), located along the y-axis on the complex plane.

Although imaginary numbers do not exist on the traditional number line, they are crucial for solving equations where solutions might not be real numbers.

Real Numbers

Real numbers are the numbers that we are most familiar with in everyday math, like integers, fractions, and decimals.
They can be positive, negative, or zero, and exist on the real number line.
In the context of complex numbers, the real number component is the part not multiplied by the imaginary unit \(i\).

• Real Part (a): For a complex number in the form \(a + bi\), \(a\) is the real number.
• Real Component in Exercise: In the exercise example, the real part is \(\frac{1}{4\sqrt{2}}\), and it lies along the x-axis in the complex plane.

Real numbers are key in calculations involving distance or magnitude, and they ensure that complex numbers can be represented graphically with a real axis.